Multilevel Monte Carlo is a key tool for approximating integrals involving expensive scientific models. The idea is to use approximations of the integrand to construct an estimator with improved accuracy over classical Monte Carlo. We propose to further enhance multilevel Monte Carlo through Bayesian surrogate models of the integrand, focusing on Gaussian process models and the associated Bayesian quadrature estimators. We show, using both theory and numerical experiments, that our approach can lead to significant improvements in accuracy when the integrand is expensive and smooth, and when the dimensionality is small or moderate. We conclude the paper with a case study illustrating the potential impact of our method in landslide-generated tsunami modelling, where the cost of each integrand evaluation is typically too large for operational settings.

翻译：多层蒙特卡洛是近似涉及昂贵科学模型积分的关键工具，其核心思想是利用被积函数的近似构造比经典蒙特卡洛精度更高的估计量。我们提出通过被积函数的贝叶斯代理模型进一步改进多层蒙特卡洛方法，重点研究高斯过程模型及其相关的贝叶斯求积估计量。理论分析与数值实验均表明，当被积函数计算成本高昂且具有光滑性，且问题维度较低或适中时，我们的方法能够显著提升精度。最后，我们通过滑坡诱发海啸建模的案例研究，展示了该方法在典型计算成本远超实际运行场景中的潜在应用价值。